Em01-G-v5 Supported-Sector Asymptotics and an Isolated Non-Tip Failure Certificate The Constant-Weight Einstein–Scalar Family

We study a fixed-profile regularization of a frozen seven-row Einstein–scalar distributional family. The complete supported response sector has an expansion with three possible inverse powers, a degree-zero logarithm, and a finite distribution, with a compact-uniform $O(\delta (1+|\log \delta |))$ remainder. Its proof uses an exact whole-support radial reduction, including the massive causal tail, and an averaged resolvent estimate with explicit parameter seminorms.

The new coefficient calculation concerns the non-tip positive $K_5$ placement. Derivative placement reduces the possible crossed order of both $47$ and positive $K_5$ to three, and their inverse-cubic coefficients vanish by an explicit tensor contraction. For positive $K_5$, a Klein–Gordon identity further reduces the exact crossed order to two. We evaluate its full inverse-quadratic coefficient, including the mass vertex, and construct a compact joint test detecting it.

On a noncollinear positive-$K_5$ incidence, support geometry excludes all six other rows. The detecting test therefore gives a complete-seven-row nonconvergence certificate for a nonempty class of admissible fixed data, without residual-tip matching. This is not an all-data classification or a global complete-family expansion. The general output-plateau supported inverse-quadratic tensor sum, the aligned negative placement, and tip matching remain open.

release{ The principal new result is an evaluated, geometrically isolated inverse-quadratic coefficient and a fixed-test failure certificate for the complete chosen regularization under explicit hypotheses.

The supported theorem is a separate sector theorem. Its lower output-plateau tensor inventory is not represented as evaluated. No physical subtraction, profile-independent normalization, gauge-invariant completion, or quantum admission is inferred.}

Figure fig:scope-Em01-G records that separation. Figure fig:split-Em01-G records one view of the same separation. Figure fig:chain-Em01-G records one view of the same separation. Figure fig:pair-Em01-G records one view of the same separation. Figure fig:gate-Em01-G records one view of the same separation. The drawings add no hypothesis.

Let $M={\mathbb{R}}^{1+3}$, with

Spacetime index conversions use $\eta$. Distributional pairings are complex bilinear, with the displayed densities.

Fix

a relatively compact separated coordinate patch $U\subset M^4$, and

Every datum is fixed before $\delta \downarrow 0$. In particular $F$ remains the advanced root of $f$.

The positive-scale family is

Its arrays, source, kernels, and vertex are defined in Appendices app:inputs–app:positive. Set

The weights are identically one and there is no $1/\hslash$ response prefactor.

Tests belong to $\mathcal{D}(U;S^{\ast })$, where $S={\mathrm{Sym}}^2(({\mathbb{R}}^4{)}^{\ast })$. For $L⋐U$, write

Compactly supported tests are extended by zero when a coordinate expression requires a fixed slightly larger neighborhood.

There are distributions

such that

and

The four singular coefficients are supported on $N_{\mathrm{mark}}$. They are constructed by (eq:scalar-coefficients) and (eq:supported-coefficients). The inverse-cubic coefficient is (eq:supported-leading).

The marked set $N_{\mathrm{mark}}$ and the moments $M_2,M_3$ are defined below.

Suppose $U$ contains a marked positive-$K_5$ incidence

with $0<\alpha <\beta$, $ϵ>0$, and $\omega \ne \mathbf{n}$. Assume $f(b)g(y)\ne 0$ and $M_2(\mathbf{n})\ne 0$ at that incidence.

There is an isolated neighborhood $O⋐U$ and a fixed ${\Phi }_{\ast }\in \mathcal{D}(O;S^{\ast })$ for which

Consequently the complete family has no raw distributional limit on $U$.

A nonempty class of admissible normalized real profiles satisfies $M_2(\mathbf{n})>0$ for every $\mathbf{n}$; it is given in Corollary cor:profile-class.

The proof, including the evaluated detecting coefficient, is in Sections sec:positive-evaluation–sec:witness. The theorem is conditional on explicit fixed data, not a universal failure statement.

“Intrinsic” means uniquely determined by the fixed positive-scale family, including its profile and scale convention. Three questions are kept separate:

1. admissible relative-cutoff independence;

2. independence of auxiliary core or residual localizers;

3. profile independence.

The first is exact at sufficiently small scales. Whole-support sector coefficients do not use an auxiliary core cutoff. A fixed residual localization defines its own localized family. Profile independence is not asserted.

On a sphere subchart choose

with

Use

On $\overline{U}$, assume positive lower bounds on $\alpha ,\beta -\alpha ,\beta -u$, and strict order and separation

The inverse is

The cone functions and density are

The two null-coordinate Jacobians have absolute determinant two.

Set

Write $\mathrm{d}\tau =\mathrm{d}\vartheta \mathrm{d}v{\mathrm{d}}^2\xi$.

For a residual multiindex $\zeta$, define

The complete supported sector has a unique representation

For the ordered transformed seed operator

the coefficients are

The transpose uses $\mathrm{d}\tau$.

Proof. On the separated future patch, $R(b-a)={\delta }_{\mathrm{D}}(u)/(8\pi \beta )$. Thus

The supported seed operator has order at most three. Restrict its ordered coefficients and transpose tangential derivatives to obtain (eq:normal-extraction). A test $r^{\zeta }{u}^j\omega (r,u)\psi (\tau )/(\zeta !j!)$, with $\omega =1$ near zero, isolates a selected normal coefficient. This proves uniqueness.

Set $h_{\mu }={\ell }_{\mu }/2$ and $\mathfrak{m}=\beta /(2\pi )$. The source symbol has its first momentum in the $G$-slot:

For constant $H$, define

The printed source obeys

Also

Proof. Consider the raised Einstein density

The mixed coefficient of $J(\eta +sH+tG)$, lowered with $\eta$, is (eq:source). The connection and Ricci mixed coefficients are $D,U$; the scalar and Einstein coefficients are $s,E$. The determinant derivative supplies the trace terms, and the two inverse-metric derivatives supply the four index-conversion terms.

The density Bianchi identity

gives (eq:mixed-divergence) at mixed order. At $p=0$, its right side vanishes.

For a direct constant-background check, the linearized Ricci symbol at $g_t=\eta +tG$ is

Its Einstein contraction is transverse. For $H=q\otimes t+t\otimes q$, the symbol is zero at every $g_t$, as is seen by substitution; the corresponding vector amplitude is ${g_t}^{-1}t$. Differentiate the raised density in $t$ to obtain (eq:constant-slot) and (eq:gauge-slot).

These are classical source identities, not quantum Ward identities.

The degree-three normal polynomial is

In particular ${c_{0,3}}^{I,J}=0$, and there are no pure residual degree-three jets.

Proof. The highest normal symbols of the response seed fields are

The order-three insertion sign is negative. The identity ${\mathsf{A}}^{\rho \mu }{}_L{\mathsf{P}}^L{}_M=-(e_M{)}^{\rho \mu }$ cancels it. The $K_8$ multiplier is $s_{\rho }/2=\beta {\ell }_{\rho }/2$. The other degree-three row is the derivative-root row. This gives (eq:top). Set $z=0$ and use (eq:constant-slot); the factor $w$ excludes pure residual terms.

Thus $j\le 2$ can be used in the supported normal form.

The pure $u^2$ coefficient is

The $K_8$ contribution is zero.

Proof. In Cartesian response coordinates $(c,s,r,y)$, all row weights are independent of $r$. The subsequent coordinate change does not mix $r$ with other coordinates. Residual order zero therefore uses only source terms with first source multiindex zero.

Let ${\mathcal{Q}}^{\mu \nu }({\partial }_s)={\mathsf{Q}}^{I,J}(0,{\partial }_s{)}^{\mu \nu }$. Its divergence in either output index is zero. The residual-order-zero $K_8$ part is

The derivative-root part reduces to

Since $\mathsf{b}(\mathsf{P}Q)=-{\mathrm{tr}}_{\eta }Q/2$, the mass-root part is

The density part is

All remaining propagation expressions have order at most two. Their pure second-normal symbol, times $\mathfrak{m}$, is (eq:c02).

One has

Proof. Contract with an arbitrary $t_{\nu }$. Completeness replaces the first input by $H=({\ell }_{♭}\otimes t+t\otimes {\ell }_{♭})/2$. Equation (eq:gauge-slot) kills both source terms in (eq:c02). Direct substitution in the density array gives

This proves (eq:projection0).

For the mixed coefficient, set $w=1$ in (eq:top). The total source momentum is $K=-h$, and $T_H(K)=0$. As $z\to 0$,

The mixed divergence identity makes the contraction with $K$ of the source $O(|z{|}^2)$. Its linear term is zero, proving (eq:projection1).

On the actual relative time region,

For a compact smooth profile $\nu$, let ${d_{\delta }}^{\nu }={\nu }_{\delta }\ast D$ locally. Admissible relative cutoffs are eventually irrelevant.

Fix $\alpha >0$, $d=\beta -\alpha >0$, and the frame. For a compact amplitude $A(v,\xi )$, supported in the separated time region, set

Compact smooth spectator dependence is allowed.

Write the smoothing displacement as

The factor four is the joint frame Jacobian. Below $\mathrm{d}H$ is frame-coordinate measure.

Define

All denominators have fixed positive lower bounds on the compact parameter ranges under consideration.

Set

where ${\omega }_{\phi }=(\cos \phi ,\sin \phi )$. Let

and

At sufficiently small positive scales,

The right side is an averaged distributional pairing. The radial amplitudes are smooth up to $t=0$, have common compact radial support, and satisfy uniform finite-order estimates.

Proof. The mode conventions in Appendix app:inputs give, on the separated relative region,

The series for $h_m$ obeys $4z{h}_m^{\prime \prime }+8h_m^{\prime }+m^2{h}_m=0$ and $4h_m(0)=-m^2/(2\pi )$, matching the causal normalization. At $m=0$, it vanishes identically.

For the cone part,

Its delta sets

Completion of the square gives

The negative multiplier changes $-i0$ to $+i0$. Before angular integration the coefficient is $-1/(32{\pi }^3{e}_{\delta })$. The substitution $\xi -\delta Z_{\delta }=\sqrt{t}{\omega }_{\phi }$, with ${\mathrm{d}}^2\xi =\frac{1}{2}\mathrm{d}t\mathrm{d}\phi$, gives the cone term of (eq:radial).

For the tail, put $z=q_{2,\delta }\ge 0$. Then

and

The tail integral is

Change to $w=t+c_{\delta }z$. The inner Jacobian is $1/c_{\delta }$, and the ratio of tail and cone prefactors is $2\pi$. This yields (eq:radial-amplitude).

Circular averaging eliminates odd Cartesian Taylor terms and is smooth in the squared radius. A radial derivative costs at most two Cartesian amplitude derivatives. The corresponding integral remainders are uniform on compact parameter sets.

Compact amplitude support bounds $\xi ,v$. The displayed substitutions bound both nonnegative radial variables. Thus the collapsed amplitude has common compact support.

For the distributional justification, replace $i0$ by $i\epsilon$, perform the ordinary changes, pair in $H$, and let $\epsilon \downarrow 0$. The map $H\mapsto L_{\delta }$ is a submersion in $B_2-B_1$. The averaged boundary values therefore exist without assigning an individual-displacement crossed product.

The integral covers the whole fixed physical support. In particular the rescaled tail $t/\delta \to \infty$ is included.

For $F\in {C_c}^{\infty }([0,\infty ))$, put $f_j=F^{(j)}(0)/j!$, and define

The sum is empty for $j=0$.

On fixed sets $0\le t\le T$, $H\in K$, $\theta \in P$, $|\delta |\le {\delta }_0$, define

All derivative orders are nonnegative. For a compact smooth weight $w$, define

The additional $r$ in the $H$-order is essential: spectator differentiation of a transformed smoothing coordinate can differentiate the original weight or amplitude in $H$.

Use

For the radial parameters (eq:radial-parameters), a smooth common-support family $F_{\delta ,H,\theta }$, and a smooth compact weight $w_{\delta ,\theta }(H)$, one has after pairing in $H$

The averaged remainder, with spectator derivatives through order $r$, is bounded by

Constants depend on fixed supports and bounded parameter derivatives of the coordinate changes, including a positive lower bound on ${\kappa }_{\delta }$.

Proof. Choose a fixed compact cutoff ${\omega }_0=1$ near zero. Write

The quotient $G$ is smooth, has common compact support, and $\Vert G{\Vert }_{C^1}\le C_N\Vert F{\Vert }_{C^{N+2}}$, also after the displayed parameter derivatives.

Polynomial division gives

Integrate explicitly where ${\omega }_0=1$, and expand the denominator away from zero. This gives

The remainder after removal of the logarithm is smooth near $z=0$. Combining these identities with $z=\delta L_{\delta }$ gives the coefficients in (eq:resolvent-expansion).

To estimate the remaining resolvent, choose $C_0>\sup |L_{\delta }|$, and a smooth ${\chi }_{\mathrm{in}}$ equal one on $[0,2C_0]$, zero on $[4C_0,\infty )$. Split by ${\chi }_{\mathrm{in}}(t/\delta )$.

On the outer support, $|t-\delta L_{\delta }|\ge t/2$, so the integral is $O(1+|\log \delta |)$ in a zeroth-order norm.

For the inner part put $t=\delta x$, and let

The explicit coordinate change is

The quantities $D_{\delta },{\kappa }_{\delta }$ are independent of $B_1,B_2$. The transformed inner test is

It has common compact support. The chain rule gives uniformly bounded first $L$-derivatives. Spectator derivatives of order $r$ can add at most $r$$H$-derivatives. These are covered by (eq:parameter-seminorm) and ${\mathcal{W}}_{N,r}$.

Uniformly for compact $x$-ranges,

Indeed, subtract $\psi (x)$ times a fixed cutoff in the principal-value integral; the remaining numerator is controlled by a first derivative. The delta part is controlled by $|\psi (x)|$. Integrating in $x,\eta$ gives an $O(1)$ inner bound.

The polynomial-division remainder contains ${\delta }^{N+1}$. Combining both pieces proves (eq:resolvent-bound). To differentiate with respect to spectators, use the $L$-coordinate representation first. Its singular factor is fixed; differentiation acts only on the transformed smooth test. This also justifies differentiation through the pairing.

For the diagnostic amplitudes ${\omega }_0,t{\omega }_0,t^2{\omega }_0$, the exact polynomial-division identities place the first logarithms at orders $0,1,2$, respectively. For amplitudes vanishing near zero, ordinary denominator expansion recovers the same finite-part coefficients and no logarithms. These checks hold for weights crossing $L=0$.

Let $f_{j,\delta ,H}={{\partial }_t}^j{\mathcal{F}}_{\delta ,H}(0)/j!$. Define

For every $N\ge 0$,

with